Irregular quadrilateral

Question

I have an irregular quadrilateral. I know the length of three sides (a, b and c) and the length of the two diagonals (e and f). All angles are unknown How do I calculate the length of the 4th side (d)?

Thank you for your help. Regards,

Mo

Answer 1

Hint : You could try re-arranging the Cosine Rule: $a^2 = c^2+b^2-2bc\cos A$ to try and find some of the angles of the triangles.

Answer 2

You could use Brahmagupta's formula which is $\sqrt{s(s-a)(s-b)(s-c)(s-d)}$ where $s$ is the semiperimeter $(\frac{1}{2})(a+b+c+d)$ only if the quadrilateral is cyclic, in which $e$ and $f$ are equal.

The proof uses angles and the law of cosines, so the side lengths a quadrilateral with no constraints on its angles cannot be determined. Take this quadrilateral and this quadrilateral which have all four side lengths measuring $\sqrt2$ units, which is another piece of evidence supporting my proof.